Use a Formula for an Arithmetic Sequence

We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.

$a_n=a_1+d(n−1)$

To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.

The common difference is $−50$, so the sequence represents a linear function with a slope of $−50$. To find the $y$ -intercept, we subtract $−50$ from $200:200−(−50)=200+50=250$. You can also find the $y$ -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in Figure 4.

Figure 4

Recall the slope-intercept form of a line is $y=mx+b$. When dealing with sequences, we use $a_n$ in place of $y$ and $n$ in place of $x$. If we know the slope and vertical intercept of the function, we can substitute them for $m$ and $b$ in the slope-intercept form of a line. Substituting $−50$ for the slope and $250$ for the vertical intercept, we get the following equation:

$a_n=−50n+250$

We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is $a_n=200−50(n−1)$, which simplifies to $a_n=−50n+250$.